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Theorem mblfinlem4 33762
Description: Backward direction of ismblfin 33763. (Contributed by Brendan Leahy, 28-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
Assertion
Ref Expression
mblfinlem4 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ))
Distinct variable group:   𝑦,𝑏,𝐴

Proof of Theorem mblfinlem4
Dummy variables 𝑓 𝑔 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltso 10403 . . . 4 < Or ℝ
21a1i 11 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → < Or ℝ)
3 simplr 776 . . 3 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) ∈ ℝ)
4 vex 3394 . . . . . . 7 𝑢 ∈ V
5 eqeq1 2810 . . . . . . . . 9 (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏)))
65anbi2d 616 . . . . . . . 8 (𝑦 = 𝑢 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑢 = (vol‘𝑏))))
76rexbidv 3240 . . . . . . 7 (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑢 = (vol‘𝑏))))
84, 7elab 3545 . . . . . 6 (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑢 = (vol‘𝑏)))
9 simprl 778 . . . . . . . . 9 ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏))) → 𝑏𝐴)
10 ovolss 23466 . . . . . . . . . . 11 ((𝑏𝐴𝐴 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝐴))
11 sstr 3806 . . . . . . . . . . . . 13 ((𝑏𝐴𝐴 ⊆ ℝ) → 𝑏 ⊆ ℝ)
12 ovolcl 23459 . . . . . . . . . . . . 13 (𝑏 ⊆ ℝ → (vol*‘𝑏) ∈ ℝ*)
1311, 12syl 17 . . . . . . . . . . . 12 ((𝑏𝐴𝐴 ⊆ ℝ) → (vol*‘𝑏) ∈ ℝ*)
14 ovolcl 23459 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
1514adantl 469 . . . . . . . . . . . 12 ((𝑏𝐴𝐴 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
16 xrlenlt 10388 . . . . . . . . . . . 12 (((vol*‘𝑏) ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝑏) ≤ (vol*‘𝐴) ↔ ¬ (vol*‘𝐴) < (vol*‘𝑏)))
1713, 15, 16syl2anc 575 . . . . . . . . . . 11 ((𝑏𝐴𝐴 ⊆ ℝ) → ((vol*‘𝑏) ≤ (vol*‘𝐴) ↔ ¬ (vol*‘𝐴) < (vol*‘𝑏)))
1810, 17mpbid 223 . . . . . . . . . 10 ((𝑏𝐴𝐴 ⊆ ℝ) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
1918ancoms 448 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ 𝑏𝐴) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
209, 19sylan2 582 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
21 simprrr 791 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → 𝑢 = (vol‘𝑏))
22 uniretop 22779 . . . . . . . . . . . . . . 15 ℝ = (topGen‘ran (,))
2322cldss 21047 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
24 dfss4 4060 . . . . . . . . . . . . . 14 (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
2523, 24sylib 209 . . . . . . . . . . . . 13 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
26 rembl 23521 . . . . . . . . . . . . . 14 ℝ ∈ dom vol
2722cldopn 21049 . . . . . . . . . . . . . . 15 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
28 opnmbl 23583 . . . . . . . . . . . . . . 15 ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
2927, 28syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
30 difmbl 23524 . . . . . . . . . . . . . 14 ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3126, 29, 30sylancr 577 . . . . . . . . . . . . 13 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
3225, 31eqeltrrd 2886 . . . . . . . . . . . 12 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
33 mblvol 23511 . . . . . . . . . . . 12 (𝑏 ∈ dom vol → (vol‘𝑏) = (vol*‘𝑏))
3432, 33syl 17 . . . . . . . . . . 11 (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏))
3534ad2antrl 710 . . . . . . . . . 10 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → (vol‘𝑏) = (vol*‘𝑏))
3621, 35eqtrd 2840 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → 𝑢 = (vol*‘𝑏))
3736breq2d 4856 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → ((vol*‘𝐴) < 𝑢 ↔ (vol*‘𝐴) < (vol*‘𝑏)))
3820, 37mtbird 316 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏𝐴𝑢 = (vol‘𝑏)))) → ¬ (vol*‘𝐴) < 𝑢)
3938rexlimdvaa 3220 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑢 = (vol‘𝑏)) → ¬ (vol*‘𝐴) < 𝑢))
408, 39syl5bi 233 . . . . 5 (𝐴 ⊆ ℝ → (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} → ¬ (vol*‘𝐴) < 𝑢))
4140ad2antrr 708 . . . 4 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))} → ¬ (vol*‘𝐴) < 𝑢))
4241imp 395 . . 3 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}) → ¬ (vol*‘𝐴) < 𝑢)
43 1rp 12050 . . . . . . . . 9 1 ∈ ℝ+
44 eqid 2806 . . . . . . . . . 10 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
4544ovolgelb 23461 . . . . . . . . 9 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)))
4643, 45mp3an3 1567 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)))
47 elmapi 8114 . . . . . . . . . . 11 (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
48 ssid 3820 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)
4944ovollb 23460 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑓) ⊆ ran ((,) ∘ 𝑓)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
5048, 49mpan2 674 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
5150adantl 469 . . . . . . . . . . . . 13 (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
52 eqid 2806 . . . . . . . . . . . . . . . 16 ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
5352, 44ovolsf 23453 . . . . . . . . . . . . . . 15 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
54 frn 6262 . . . . . . . . . . . . . . . 16 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
55 icossxr 12476 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℝ*
5654, 55syl6ss 3810 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
57 supxrcl 12363 . . . . . . . . . . . . . . 15 (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
5853, 56, 573syl 18 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
59 peano2re 10494 . . . . . . . . . . . . . . 15 ((vol*‘𝐴) ∈ ℝ → ((vol*‘𝐴) + 1) ∈ ℝ)
6059rexrd 10374 . . . . . . . . . . . . . 14 ((vol*‘𝐴) ∈ ℝ → ((vol*‘𝐴) + 1) ∈ ℝ*)
61 rncoss 5587 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) ⊆ ran (,)
6261unissi 4655 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑓) ⊆ ran (,)
63 unirnioo 12492 . . . . . . . . . . . . . . . . 17 ℝ = ran (,)
6462, 63sseqtr4i 3835 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑓) ⊆ ℝ
65 ovolcl 23459 . . . . . . . . . . . . . . . 16 ( ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*)
6664, 65ax-mp 5 . . . . . . . . . . . . . . 15 (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ*
67 xrletr 12207 . . . . . . . . . . . . . . 15 (((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
6866, 67mp3an1 1565 . . . . . . . . . . . . . 14 ((sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
6958, 60, 68syl2anr 586 . . . . . . . . . . . . 13 (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (((vol*‘ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7051, 69mpand 678 . . . . . . . . . . . 12 (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7170adantll 696 . . . . . . . . . . 11 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7247, 71sylan2 582 . . . . . . . . . 10 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7372anim2d 601 . . . . . . . . 9 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)) → ((𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))))
7473reximdva 3204 . . . . . . . 8 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))))
7546, 74mpd 15 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
76 rexex 3189 . . . . . . 7 (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → ∃𝑓(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7775, 76syl 17 . . . . . 6 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
7877ad2antrr 708 . . . . 5 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑓(𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
79 difss 3936 . . . . . . . . . . . . . 14 ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑓)
8079, 64sstri 3807 . . . . . . . . . . . . 13 ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ
81 ovolcl 23459 . . . . . . . . . . . . 13 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ*)
8280, 81ax-mp 5 . . . . . . . . . . . 12 (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ*
8359, 82jctil 511 . . . . . . . . . . 11 ((vol*‘𝐴) ∈ ℝ → ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
8483ad4antlr 717 . . . . . . . . . 10 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
85 ovolss 23466 . . . . . . . . . . . . . . 15 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘ ran ((,) ∘ 𝑓)))
8679, 64, 85mp2an 675 . . . . . . . . . . . . . 14 (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘ ran ((,) ∘ 𝑓))
87 xrletr 12207 . . . . . . . . . . . . . . . 16 (((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
8882, 66, 87mp3an12 1568 . . . . . . . . . . . . . . 15 (((vol*‘𝐴) + 1) ∈ ℝ* → (((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
8960, 88syl 17 . . . . . . . . . . . . . 14 ((vol*‘𝐴) ∈ ℝ → (((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
9086, 89mpani 679 . . . . . . . . . . . . 13 ((vol*‘𝐴) ∈ ℝ → ((vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
9190ad4antlr 717 . . . . . . . . . . . 12 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ 𝐴 ran ((,) ∘ 𝑓)) → ((vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
9291impr 444 . . . . . . . . . . 11 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1))
93 ovolge0 23462 . . . . . . . . . . . 12 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ → 0 ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
9480, 93ax-mp 5 . . . . . . . . . . 11 0 ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))
9592, 94jctil 511 . . . . . . . . . 10 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (0 ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
96 xrrege0 12223 . . . . . . . . . 10 ((((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ) ∧ (0 ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
9784, 95, 96syl2anc 575 . . . . . . . . 9 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
98 resubcl 10630 . . . . . . . . . . . . . 14 (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
9998adantrr 699 . . . . . . . . . . . . 13 (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
100 posdif 10806 . . . . . . . . . . . . . . . 16 ((𝑢 ∈ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝑢 < (vol*‘𝐴) ↔ 0 < ((vol*‘𝐴) − 𝑢)))
101100ancoms 448 . . . . . . . . . . . . . . 15 (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘𝐴) ↔ 0 < ((vol*‘𝐴) − 𝑢)))
102101biimpd 220 . . . . . . . . . . . . . 14 (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘𝐴) → 0 < ((vol*‘𝐴) − 𝑢)))
103102impr 444 . . . . . . . . . . . . 13 (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → 0 < ((vol*‘𝐴) − 𝑢))
10499, 103elrpd 12083 . . . . . . . . . . . 12 (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ+)
105104rphalfcld 12098 . . . . . . . . . . 11 (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
1063, 105sylan 571 . . . . . . . . . 10 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
107106adantr 468 . . . . . . . . 9 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
108 eqid 2806 . . . . . . . . . . 11 seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
109108ovolgelb 23461 . . . . . . . . . 10 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ ∧ (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
11080, 109mp3an1 1565 . . . . . . . . 9 (((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ ∧ (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
11197, 107, 110syl2anc 575 . . . . . . . 8 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
112 elmapi 8114 . . . . . . . . . . 11 (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ) → 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
113 ssid 3820 . . . . . . . . . . . . . 14 ran ((,) ∘ 𝑔) ⊆ ran ((,) ∘ 𝑔)
114108ovollb 23460 . . . . . . . . . . . . . 14 ((𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ran ((,) ∘ 𝑔) ⊆ ran ((,) ∘ 𝑔)) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
115113, 114mpan2 674 . . . . . . . . . . . . 13 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
116115adantl 469 . . . . . . . . . . . 12 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
117 eqid 2806 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
118117, 108ovolsf 23453 . . . . . . . . . . . . . 14 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞))
119 frn 6262 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ (0[,)+∞))
120119, 55syl6ss 3810 . . . . . . . . . . . . . 14 (seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ ℝ*)
121 supxrcl 12363 . . . . . . . . . . . . . 14 (ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
122118, 120, 1213syl 18 . . . . . . . . . . . . 13 (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
12399rehalfcld 11546 . . . . . . . . . . . . . . . . 17 (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
1243, 123sylan 571 . . . . . . . . . . . . . . . 16 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
125124adantr 468 . . . . . . . . . . . . . . 15 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
12697, 125readdcld 10354 . . . . . . . . . . . . . 14 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ)
127126rexrd 10374 . . . . . . . . . . . . 13 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*)
128 rncoss 5587 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑔) ⊆ ran (,)
129128unissi 4655 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑔) ⊆ ran (,)
130129, 63sseqtr4i 3835 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑔) ⊆ ℝ
131 ovolcl 23459 . . . . . . . . . . . . . . 15 ( ran ((,) ∘ 𝑔) ⊆ ℝ → (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ*)
132130, 131ax-mp 5 . . . . . . . . . . . . . 14 (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ*
133 xrletr 12207 . . . . . . . . . . . . . 14 (((vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
134132, 133mp3an1 1565 . . . . . . . . . . . . 13 ((sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*) → (((vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
135122, 127, 134syl2anr 586 . . . . . . . . . . . 12 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (((vol*‘ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
136116, 135mpand 678 . . . . . . . . . . 11 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
137112, 136sylan2 582 . . . . . . . . . 10 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
138137anim2d 601 . . . . . . . . 9 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)) → ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))))
139138reximdva 3204 . . . . . . . 8 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))))
140111, 139mpd 15 . . . . . . 7 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
141 rexex 3189 . . . . . . 7 (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → ∃𝑔(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
142140, 141syl 17 . . . . . 6 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔(( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
14359, 66jctil 511 . . . . . . . . . . . 12 ((vol*‘𝐴) ∈ ℝ → ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
144143ad3antlr 713 . . . . . . . . . . 11 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
145 ovolge0 23462 . . . . . . . . . . . . . 14 ( ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑓)))
14664, 145ax-mp 5 . . . . . . . . . . . . 13 0 ≤ (vol*‘ ran ((,) ∘ 𝑓))
147146jctl 515 . . . . . . . . . . . 12 ((vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
148147adantl 469 . . . . . . . . . . 11 ((𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
149 xrrege0 12223 . . . . . . . . . . 11 ((((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ) ∧ (0 ≤ (vol*‘ ran ((,) ∘ 𝑓)) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
150144, 148, 149syl2an 585 . . . . . . . . . 10 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
151150, 125resubcld 10743 . . . . . . . . 9 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ)
152150, 107ltsubrpd 12118 . . . . . . . . 9 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘ ran ((,) ∘ 𝑓)))
153 retop 22778 . . . . . . . . . . 11 (topGen‘ran (,)) ∈ Top
154 retopbas 22777 . . . . . . . . . . . . 13 ran (,) ∈ TopBases
155 bastg 20984 . . . . . . . . . . . . 13 (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
156154, 155ax-mp 5 . . . . . . . . . . . 12 ran (,) ⊆ (topGen‘ran (,))
15761, 156sstri 3807 . . . . . . . . . . 11 ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
158 uniopn 20915 . . . . . . . . . . 11 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
159153, 157, 158mp2an 675 . . . . . . . . . 10 ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
160 mblfinlem2 33760 . . . . . . . . . 10 (( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘ ran ((,) ∘ 𝑓))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
161159, 160mp3an1 1565 . . . . . . . . 9 ((((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘ ran ((,) ∘ 𝑓))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
162151, 152, 161syl2anc 575 . . . . . . . 8 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
163162adantr 468 . . . . . . 7 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
164 indif2 4072 . . . . . . . . . . . . . . 15 (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑔))) = ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑔))
16522cldss 21047 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ)
166 df-ss 3783 . . . . . . . . . . . . . . . . 17 (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
167165, 166sylib 209 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ ℝ) = 𝑠)
168167difeq1d 3926 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑠 ∩ ℝ) ∖ ran ((,) ∘ 𝑔)) = (𝑠 ran ((,) ∘ 𝑔)))
169164, 168syl5eq 2852 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑔))) = (𝑠 ran ((,) ∘ 𝑔)))
170128, 156sstri 3807 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑔) ⊆ (topGen‘ran (,))
171 uniopn 20915 . . . . . . . . . . . . . . . . 17 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑔) ⊆ (topGen‘ran (,))) → ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,)))
172153, 170, 171mp2an 675 . . . . . . . . . . . . . . . 16 ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,))
17322opncld 21051 . . . . . . . . . . . . . . . 16 (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,))) → (ℝ ∖ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))))
174153, 172, 173mp2an 675 . . . . . . . . . . . . . . 15 (ℝ ∖ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,)))
175 incld 21061 . . . . . . . . . . . . . . 15 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑔))) ∈ (Clsd‘(topGen‘ran (,))))
176174, 175mpan2 674 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ran ((,) ∘ 𝑔))) ∈ (Clsd‘(topGen‘ran (,))))
177169, 176eqeltrrd 2886 . . . . . . . . . . . . 13 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))))
178 simpr 473 . . . . . . . . . . . . . . 15 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → 𝑠 ran ((,) ∘ 𝑓))
179 simpl 470 . . . . . . . . . . . . . . 15 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → ( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔))
180178, 179ssdif2d 3948 . . . . . . . . . . . . . 14 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → (𝑠 ran ((,) ∘ 𝑔)) ⊆ ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))
181 dfin4 4069 . . . . . . . . . . . . . . 15 ( ran ((,) ∘ 𝑓) ∩ 𝐴) = ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))
182 inss2 4030 . . . . . . . . . . . . . . 15 ( ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ 𝐴
183181, 182eqsstr3i 3833 . . . . . . . . . . . . . 14 ( ran ((,) ∘ 𝑓) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ 𝐴
184180, 183syl6ss 3810 . . . . . . . . . . . . 13 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → (𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴)
185 sseq1 3823 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 ran ((,) ∘ 𝑔)) → (𝑏𝐴 ↔ (𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴))
186185anbi1d 617 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑔)) → ((𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ↔ ((𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
187 fveq2 6408 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑔)) = 𝑏 → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))
188187eqcoms 2814 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 ran ((,) ∘ 𝑔)) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))
189188biantrud 523 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 ran ((,) ∘ 𝑔)) → ((𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴 ↔ ((𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
190186, 189bitr4d 273 . . . . . . . . . . . . . 14 (𝑏 = (𝑠 ran ((,) ∘ 𝑔)) → ((𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ↔ (𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴))
191190rspcev 3502 . . . . . . . . . . . . 13 (((𝑠 ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
192177, 184, 191syl2an 585 . . . . . . . . . . . 12 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
193192an12s 631 . . . . . . . . . . 11 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑠 ran ((,) ∘ 𝑓))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
194193adantrrr 707 . . . . . . . . . 10 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
195194adantlr 697 . . . . . . . . 9 (((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
196195adantll 696 . . . . . . . 8 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
197 difss 3936 . . . . . . . . . . . 12 (𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))) ⊆ 𝐴
198 ovolsscl 23467 . . . . . . . . . . . 12 (((𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℝ)
199197, 198mp3an1 1565 . . . . . . . . . . 11 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℝ)
200199ad5antr 719 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℝ)
201 simp-6r 802 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘𝐴) ∈ ℝ)
202 simpl 470 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴)) → 𝑢 ∈ ℝ)
203202ad4antlr 717 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 ∈ ℝ)
204 difdif2 4086 . . . . . . . . . . . 12 (𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))) = ((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))
205204fveq2i 6411 . . . . . . . . . . 11 (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) = (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔))))
206 difss 3936 . . . . . . . . . . . . . . 15 (𝐴𝑠) ⊆ 𝐴
207 inss1 4029 . . . . . . . . . . . . . . 15 (𝐴 ran ((,) ∘ 𝑔)) ⊆ 𝐴
208206, 207unssi 3987 . . . . . . . . . . . . . 14 ((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔))) ⊆ 𝐴
209 ovolsscl 23467 . . . . . . . . . . . . . 14 ((((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ∈ ℝ)
210208, 209mp3an1 1565 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ∈ ℝ)
211210ad5antr 719 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ∈ ℝ)
212 ovolsscl 23467 . . . . . . . . . . . . . . 15 (((𝐴𝑠) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
213206, 212mp3an1 1565 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴𝑠)) ∈ ℝ)
214213ad5antr 719 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴𝑠)) ∈ ℝ)
215 ovolsscl 23467 . . . . . . . . . . . . . . 15 (((𝐴 ran ((,) ∘ 𝑔)) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ∈ ℝ)
216207, 215mp3an1 1565 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ∈ ℝ)
217216ad5antr 719 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ∈ ℝ)
218214, 217readdcld 10354 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))) ∈ ℝ)
2193, 202, 98syl2an 585 . . . . . . . . . . . . 13 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
220219ad3antrrr 712 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
221 ssdifss 3940 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℝ → (𝐴𝑠) ⊆ ℝ)
222221adantr 468 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴𝑠) ⊆ ℝ)
223 ssinss1 4038 . . . . . . . . . . . . . . 15 (𝐴 ⊆ ℝ → (𝐴 ran ((,) ∘ 𝑔)) ⊆ ℝ)
224223adantr 468 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ran ((,) ∘ 𝑔)) ⊆ ℝ)
225 ovolun 23480 . . . . . . . . . . . . . 14 ((((𝐴𝑠) ⊆ ℝ ∧ (vol*‘(𝐴𝑠)) ∈ ℝ) ∧ ((𝐴 ran ((,) ∘ 𝑔)) ⊆ ℝ ∧ (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ∈ ℝ)) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))))
226222, 213, 224, 216, 225syl22anc 858 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))))
227226ad5antr 719 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))))
228124ad2antrr 708 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
229228adantr 468 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
230150ad2antrr 708 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
231 simprl 778 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → 𝑠 ran ((,) ∘ 𝑓))
232150adantr 468 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ)
233 ovolsscl 23467 . . . . . . . . . . . . . . . . . 18 ((𝑠 ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
23464, 233mp3an2 1566 . . . . . . . . . . . . . . . . 17 ((𝑠 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
235231, 232, 234syl2anr 586 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘𝑠) ∈ ℝ)
236230, 235resubcld 10743 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)) ∈ ℝ)
237 ssdif 3944 . . . . . . . . . . . . . . . . . . 19 (𝐴 ran ((,) ∘ 𝑓) → (𝐴𝑠) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑠))
238 difss 3936 . . . . . . . . . . . . . . . . . . . 20 ( ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ran ((,) ∘ 𝑓)
239238, 64sstri 3807 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ℝ
240 ovolss 23466 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑠) ⊆ ( ran ((,) ∘ 𝑓) ∖ 𝑠) ∧ ( ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ℝ) → (vol*‘(𝐴𝑠)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)))
241237, 239, 240sylancl 576 . . . . . . . . . . . . . . . . . 18 (𝐴 ran ((,) ∘ 𝑓) → (vol*‘(𝐴𝑠)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)))
242241adantr 468 . . . . . . . . . . . . . . . . 17 ((𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘(𝐴𝑠)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)))
243242ad3antlr 713 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴𝑠)) ≤ (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)))
244 eleq1w 2868 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
245244, 32vtoclga 3465 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol)
246245adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → 𝑠 ∈ dom vol)
247 mblsplit 23513 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠))))
24864, 247mp3an2 1566 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ dom vol ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠))))
249246, 232, 248syl2anr 586 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘ ran ((,) ∘ 𝑓)) = ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠))))
250249eqcomd 2812 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠))) = (vol*‘ ran ((,) ∘ 𝑓)))
251230recnd 10353 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℂ)
252 inss1 4029 . . . . . . . . . . . . . . . . . . . . . . 23 ( ran ((,) ∘ 𝑓) ∩ 𝑠) ⊆ ran ((,) ∘ 𝑓)
253 ovolsscl 23467 . . . . . . . . . . . . . . . . . . . . . . 23 ((( ran ((,) ∘ 𝑓) ∩ 𝑠) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
254252, 64, 253mp3an12 1568 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
255150, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
256255ad2antrr 708 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
257256recnd 10353 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℂ)
258 ovolsscl 23467 . . . . . . . . . . . . . . . . . . . . . . 23 ((( ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ran ((,) ∘ 𝑓) ∧ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
259238, 64, 258mp3an12 1568 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*‘ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
260150, 259syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
261260recnd 10353 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℂ)
262261ad2antrr 708 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℂ)
263251, 257, 262subaddd 10695 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) ↔ ((vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠))) = (vol*‘ ran ((,) ∘ 𝑓))))
264250, 263mpbird 248 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)))
265 sseqin2 4016 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ran ((,) ∘ 𝑓) ↔ ( ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
266265biimpi 207 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ran ((,) ∘ 𝑓) → ( ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
267266fveq2d 6412 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ran ((,) ∘ 𝑓) → (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠)) = (vol*‘𝑠))
268267oveq2d 6890 . . . . . . . . . . . . . . . . . . 19 (𝑠 ran ((,) ∘ 𝑓) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
269268adantr 468 . . . . . . . . . . . . . . . . . 18 ((𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
270269ad2antll 711 . . . . . . . . . . . . . . . . 17 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘( ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
271264, 270eqtr3d 2842 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝑠)) = ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
272243, 271breqtrd 4870 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴𝑠)) ≤ ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
273 simprrr 791 . . . . . . . . . . . . . . . 16 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))
274230, 229, 235, 273ltsub23d 10917 . . . . . . . . . . . . . . 15 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)) < (((vol*‘𝐴) − 𝑢) / 2))
275214, 236, 229, 272, 274lelttrd 10480 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴𝑠)) < (((vol*‘𝐴) − 𝑢) / 2))
276216ad4antr 715 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ∈ ℝ)
277126, 132jctil 511 . . . . . . . . . . . . . . . . . 18 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ))
278 simpr 473 . . . . . . . . . . . . . . . . . . 19 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))
279 ovolge0 23462 . . . . . . . . . . . . . . . . . . . 20 ( ran ((,) ∘ 𝑔) ⊆ ℝ → 0 ≤ (vol*‘ ran ((,) ∘ 𝑔)))
280130, 279ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ≤ (vol*‘ ran ((,) ∘ 𝑔))
281278, 280jctil 511 . . . . . . . . . . . . . . . . . 18 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (0 ≤ (vol*‘ ran ((,) ∘ 𝑔)) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
282 xrrege0 12223 . . . . . . . . . . . . . . . . . 18 ((((vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ) ∧ (0 ≤ (vol*‘ ran ((,) ∘ 𝑔)) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ)
283277, 281, 282syl2an 585 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ)
284 difss 3936 . . . . . . . . . . . . . . . . . 18 ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ran ((,) ∘ 𝑔)
285 ovolsscl 23467 . . . . . . . . . . . . . . . . . 18 ((( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ran ((,) ∘ 𝑔) ∧ ran ((,) ∘ 𝑔) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
286284, 130, 285mp3an12 1568 . . . . . . . . . . . . . . . . 17 ((vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ → (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
287283, 286syl 17 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
288 ssun2 3976 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑔) ∩ 𝐴) ⊆ (( ran ((,) ∘ 𝑔) ∖ ran ((,) ∘ 𝑓)) ∪ ( ran ((,) ∘ 𝑔) ∩ 𝐴))
289 incom 4004 . . . . . . . . . . . . . . . . . . 19 (𝐴 ran ((,) ∘ 𝑔)) = ( ran ((,) ∘ 𝑔) ∩ 𝐴)
290 difdif2 4086 . . . . . . . . . . . . . . . . . . 19 ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = (( ran ((,) ∘ 𝑔) ∖ ran ((,) ∘ 𝑓)) ∪ ( ran ((,) ∘ 𝑔) ∩ 𝐴))
291288, 289, 2903sstr4i 3841 . . . . . . . . . . . . . . . . . 18 (𝐴 ran ((,) ∘ 𝑔)) ⊆ ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))
292284, 130sstri 3807 . . . . . . . . . . . . . . . . . 18 ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ
293291, 292pm3.2i 458 . . . . . . . . . . . . . . . . 17 ((𝐴 ran ((,) ∘ 𝑔)) ⊆ ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ)
294 ovolss 23466 . . . . . . . . . . . . . . . . 17 (((𝐴 ran ((,) ∘ 𝑔)) ⊆ ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ ( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ≤ (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))))
295293, 294mp1i 13 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ≤ (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))))
296 opnmbl 23583 . . . . . . . . . . . . . . . . . . . . . . 23 ( ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ran ((,) ∘ 𝑓) ∈ dom vol)
297159, 296ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ran ((,) ∘ 𝑓) ∈ dom vol
298 difmbl 23524 . . . . . . . . . . . . . . . . . . . . . 22 (( ran ((,) ∘ 𝑓) ∈ dom vol ∧ 𝐴 ∈ dom vol) → ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
299297, 298mpan 673 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ dom vol → ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
300299ad4antlr 717 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ( ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
301 mblsplit 23513 . . . . . . . . . . . . . . . . . . . . 21 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol ∧ ran ((,) ∘ 𝑔) ⊆ ℝ ∧ (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑔)) = ((vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))))
302130, 301mp3an2 1566 . . . . . . . . . . . . . . . . . . . 20 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol ∧ (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘ ran ((,) ∘ 𝑔)) = ((vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))))
303300, 283, 302syl2anc 575 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑔)) = ((vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))))
304 sseqin2 4016 . . . . . . . . . . . . . . . . . . . . . . 23 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ↔ ( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = ( ran ((,) ∘ 𝑓) ∖ 𝐴))
305304biimpi 207 . . . . . . . . . . . . . . . . . . . . . 22 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) → ( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴)) = ( ran ((,) ∘ 𝑓) ∖ 𝐴))
306305fveq2d 6412 . . . . . . . . . . . . . . . . . . . . 21 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) → (vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)))
307306oveq1d 6889 . . . . . . . . . . . . . . . . . . . 20 (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) → ((vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))) = ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))))
308307ad2antrl 710 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘( ran ((,) ∘ 𝑔) ∩ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))) = ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))))
309303, 308eqtr2d 2841 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))) = (vol*‘ ran ((,) ∘ 𝑔)))
310283recnd 10353 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑔)) ∈ ℂ)
31197adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
312311recnd 10353 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℂ)
313287recnd 10353 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℂ)
314310, 312, 313subaddd 10695 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘ ran ((,) ∘ 𝑔)) − (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ↔ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴)))) = (vol*‘ ran ((,) ∘ 𝑔))))
315309, 314mpbird 248 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘ ran ((,) ∘ 𝑔)) − (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))))
316 simprr 780 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))
317283, 311, 228lesubadd2d 10911 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘ ran ((,) ∘ 𝑔)) − (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2) ↔ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
318316, 317mpbird 248 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘ ran ((,) ∘ 𝑔)) − (vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
319315, 318eqbrtrrd 4868 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘( ran ((,) ∘ 𝑔) ∖ ( ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
320276, 287, 228, 295, 319letrd 10479 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
321320adantr 468 . . . . . . . . . . . . . 14 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ran ((,) ∘ 𝑔))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
322214, 217, 229, 229, 275, 321ltleaddd 10933 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))) < ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)))
32398recnd 10353 . . . . . . . . . . . . . . . . 17 (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘𝐴) − 𝑢) ∈ ℂ)
3243232halvesd 11545 . . . . . . . . . . . . . . . 16 (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
325324adantll 696 . . . . . . . . . . . . . . 15 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ ℝ) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
326325ad2ant2r 744 . . . . . . . . . . . . . 14 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
327326ad3antrrr 712 . . . . . . . . . . . . 13 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
328322, 327breqtrd 4870 . . . . . . . . . . . 12 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴𝑠)) + (vol*‘(𝐴 ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
329211, 218, 220, 227, 328lelttrd 10480 . . . . . . . . . . 11 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴𝑠) ∪ (𝐴 ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
330205, 329syl5eqbr 4879 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
331200, 201, 203, 330ltsub13d 10918 . . . . . . . . 9 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 < ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))))
332 opnmbl 23583 . . . . . . . . . . . . . . . 16 ( ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,)) → ran ((,) ∘ 𝑔) ∈ dom vol)
333172, 332ax-mp 5 . . . . . . . . . . . . . . 15 ran ((,) ∘ 𝑔) ∈ dom vol
334 difmbl 23524 . . . . . . . . . . . . . . 15 ((𝑠 ∈ dom vol ∧ ran ((,) ∘ 𝑔) ∈ dom vol) → (𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol)
335245, 333, 334sylancl 576 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol)
336 mblvol 23511 . . . . . . . . . . . . . 14 ((𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
337335, 336syl 17 . . . . . . . . . . . . 13 (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
338337ad2antrl 710 . . . . . . . . . . . 12 (((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
339 sseqin2 4016 . . . . . . . . . . . . . . . 16 ((𝑠 ran ((,) ∘ 𝑔)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔))) = (𝑠 ran ((,) ∘ 𝑔)))
340184, 339sylib 209 . . . . . . . . . . . . . . 15 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → (𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔))) = (𝑠 ran ((,) ∘ 𝑔)))
341340fveq2d 6412 . . . . . . . . . . . . . 14 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ 𝑠 ran ((,) ∘ 𝑓)) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
342341adantrr 699 . . . . . . . . . . . . 13 ((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
343342ad2ant2rl 746 . . . . . . . . . . . 12 (((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ran ((,) ∘ 𝑔))))
344338, 343eqtr4d 2843 . . . . . . . . . . 11 (((( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))))
345344adantll 696 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))))
346335adantr 468 . . . . . . . . . . . 12 ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → (𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol)
347 simp-4l 792 . . . . . . . . . . . 12 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
348 mblsplit 23513 . . . . . . . . . . . . . 14 (((𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))))
3493483expb 1142 . . . . . . . . . . . . 13 (((𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))))
350349eqcomd 2812 . . . . . . . . . . . 12 (((𝑠 ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘𝐴))
351346, 347, 350syl2anr 586 . . . . . . . . . . 11 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘𝐴))
352 recn 10311 . . . . . . . . . . . . . 14 ((vol*‘𝐴) ∈ ℝ → (vol*‘𝐴) ∈ ℂ)
353352adantl 469 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) ∈ ℂ)
354199recnd 10353 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℂ)
355 inss1 4029 . . . . . . . . . . . . . . 15 (𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔))) ⊆ 𝐴
356 ovolsscl 23467 . . . . . . . . . . . . . . 15 (((𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔))) ⊆ 𝐴𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℝ)
357355, 356mp3an1 1565 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℝ)
358357recnd 10353 . . . . . . . . . . . . 13 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) ∈ ℂ)
359353, 354, 358subadd2d 10696 . . . . . . . . . . . 12 ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) ↔ ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘𝐴)))
360359ad5antr 719 . . . . . . . . . . 11 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) ↔ ((vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘𝐴)))
361351, 360mpbird 248 . . . . . . . . . 10 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ran ((,) ∘ 𝑔)))))
362345, 361eqtr4d 2843 . . . . . . . . 9 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ran ((,) ∘ 𝑔))) = ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ran ((,) ∘ 𝑔))))))
363331, 362breqtrrd 4872 . . . . . . . 8 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑔))))
364 fvex 6421 . . . . . . . . 9 (vol‘(𝑠 ran ((,) ∘ 𝑔))) ∈ V
365 eqeq1 2810 . . . . . . . . . . . 12 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑔))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
366365anbi2d 616 . . . . . . . . . . 11 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑔))) → ((𝑏𝐴𝑣 = (vol‘𝑏)) ↔ (𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
367366rexbidv 3240 . . . . . . . . . 10 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑔))) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
368 breq2 4848 . . . . . . . . . 10 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑔))) → (𝑢 < 𝑣𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑔)))))
369367, 368anbi12d 618 . . . . . . . . 9 (𝑣 = (vol‘(𝑠 ran ((,) ∘ 𝑔))) → ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣) ↔ (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑔))))))
370364, 369spcev 3493 . . . . . . . 8 ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴 ∧ (vol‘(𝑠 ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ran ((,) ∘ 𝑔)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
371196, 363, 370syl2anc 575 . . . . . . 7 (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ran ((,) ∘ 𝑓) ∧ ((vol*‘ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
372163, 371rexlimddv 3223 . . . . . 6 ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ (( ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ran ((,) ∘ 𝑔) ∧ (vol*‘ ran ((,) ∘ 𝑔)) ≤ ((vol*‘( ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
373142, 372exlimddv 2026 . . . . 5 (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ran ((,) ∘ 𝑓) ∧ (vol*‘ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
37478, 373exlimddv 2026 . . . 4 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
375 eqeq1 2810 . . . . . . 7 (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏)))
376375anbi2d 616 . . . . . 6 (𝑦 = 𝑣 → ((𝑏𝐴𝑦 = (vol‘𝑏)) ↔ (𝑏𝐴𝑣 = (vol‘𝑏))))
377376rexbidv 3240 . . . . 5 (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏))))
378377rexab 3565 . . . 4 (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑢 < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
379374, 378sylibr 225 . . 3 ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}𝑢 < 𝑣)
3802, 3, 42, 379eqsupd 8602 . 2 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘𝐴))
381380eqcomd 2812 1 (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏𝐴𝑦 = (vol‘𝑏))}, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wrex 3097  cdif 3766  cun 3767  cin 3768  wss 3769   cuni 4630   class class class wbr 4844   Or wor 5231   × cxp 5309  dom cdm 5311  ran crn 5312  ccom 5315  wf 6097  cfv 6101  (class class class)co 6874  𝑚 cmap 8092  supcsup 8585  cc 10219  cr 10220  0cc0 10221  1c1 10222   + caddc 10224  +∞cpnf 10356  *cxr 10358   < clt 10359  cle 10360  cmin 10551   / cdiv 10969  cn 11305  2c2 11356  +crp 12046  (,)cioo 12393  [,)cico 12395  seqcseq 13024  abscabs 14197  topGenctg 16303  Topctop 20911  TopBasesctb 20963  Clsdccld 21034  vol*covol 23443  volcvol 23444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-disj 4813  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-omul 7801  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-fi 8556  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-acn 9051  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-n0 11560  df-z 11644  df-uz 11905  df-q 12008  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-ioo 12397  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-rlim 14443  df-sum 14640  df-rest 16288  df-topgen 16309  df-psmet 19946  df-xmet 19947  df-met 19948  df-bl 19949  df-mopn 19950  df-top 20912  df-topon 20929  df-bases 20964  df-cld 21037  df-cmp 21404  df-conn 21429  df-ovol 23445  df-vol 23446
This theorem is referenced by:  ismblfin  33763
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